The Hoops had reclaimed their league title midweek with a 1-1 draw away at Dundee United, recapturing top honours from rivals Rangers who had ended their winning streak last season.
Postecoglou, who arrived at the start of the season, initially got off to a tricky start when his side exited the Champions League qualifiers, but later turned things around.
The Greek-born Australian coach captured Celtic's 52nd league title, as well as supporters' hearts, and they rewarded him with a rapturous reception back at home.
The 56-year-old seemed to be struggling to hold in his emotions as fans showed their appreciation, and returned the favour with a rousing victory speech.
"Champions, that's who we are," Postecoglu said.
"I just want to thank everyone involved, from the board management, I've got the best staff in the land, unbelievable effort.
"I've got a group of players who have been unbelievable, champions of Scotland and well deserved."
The coach then directed his appreciation to the supporters, saying: "And for you the best on the planet, unbelievable.
"You've embraced me, you've embraced our family, you've embraced this jumper, and I want everyone to enjoy today, enjoy this summer, and we'll come back bigger, better, because we never stop."